MHB Is the language ${L}_{n}$ regular?

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The discussion centers on determining whether the language ${L}_{n} = \{ a^{k} | k \text{ is a multiple of } n \}$ is regular for each positive integer n. The original poster expresses uncertainty about using the pumping lemma and seeks guidance on constructing a non-deterministic finite automaton (NFA), deterministic finite automaton (DFA), or regular expression for this language. Participants suggest that regular expressions are a straightforward way to define regular languages and recommend reviewing definitions and examples. The poster struggles with creating a regular expression for $L_1$, which is defined as $\{a^k | k \ge 0\} = \{a\}^*$. The conversation emphasizes the importance of understanding regular expressions in the context of regular languages.
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Hi, I'm back with another question, but the opposite of last time.

The question is:

For each positive integer $n$, let ${L}_{n}$ = { ${a}^{k}$ $|$ $k$ is a multiple of $n$ }
Show that for each $n$ the language ${L}_{n}$ is regular. As far as I understand you cannot use pumping lemma to prove a language is regular.

I assume that leaves me with having to do a NFA, DFA or regular expression as I don't know where to begin to create one of those for this language.

Thanks!
 
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The easiest way to define regular languages is using regular expressions. Can you write a regular expression describing $L_1=\{a^k\mid k\ge0\}=\{a\}^*$?
 
Evgeny.Makarov said:
The easiest way to define regular languages is using regular expressions. Can you write a regular expression describing $L_1=\{a^k\mid k\ge0\}=\{a\}^*$?

Honestly no, I just can't see it. I'm really struggling with this particular problem sheet.
 
Then you should read the definition and examples of regular expressions, for example, in Wikipedia.
 

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